#8622090

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#44265	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4025 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,11)(4,13)(5,12)(6,10)(7,8)(9,15)(14,16)(17,18,23,26)(19,21,24,29)(20,22,25,31)(27,28,30,32)(33,34,39,42)(35,40,45,48)(36,37,38,41)(43,44,46,47);(1,3,7,12)(2,5,8,11)(4,14,15,6)(9,16,13,10)(17,19,23,24)(18,21,26,29)(20,27,25,30)(22,28,31,32)(33,35,39,45)(34,40,42,48)(36,46,38,43)(37,47,41,44);(2,8)(5,11)(9,13)(10,16)(19,24)(21,29)(27,30)(28,32)(33,36,39,38)(34,41,42,37)(35,43,45,46)(40,47,48,44);(1,4,7,15)(2,9,8,13)(3,6,12,14)(5,10,11,16)(17,20)(18,22)(19,27)(21,28)(23,25)(24,30)(26,31)(29,32)(33,35)(34,40)(36,43)(37,44)(38,46)(39,45)(41,47)(42,48);(1,5)(2,3)(4,10)(6,9)(7,11)(8,12)(13,14)(15,16)(17,21,23,29)(18,24,26,19)(20,28,25,32)(22,30,31,27)(33,37,39,41)(34,36,42,38)(35,44,45,47)(40,43,48,46);(1,6)(2,10)(3,4)(5,9)(7,14)(8,16)(11,13)(12,15)(17,22)(18,25)(19,28)(20,26)(21,30)(23,31)(24,32)(27,29)(33,38,39,36)(34,41,42,37)(35,43,45,46)(40,44,48,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,d^2=g,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8529286);